We examine the optimal trading strategy for a CRRA investor who maximizes the expected utility of wealth on a finite date and faces transaction costs. Closed-form solutions are obtained when this date is uncertain. We then show a sequence of analytical solutions converge to the solution to the problem with a deterministic finite horizon. Consistent with the common life-cycle investment advice, the optimal trading strategy is found to be horizon dependent and largely buy and hold. Moreover, it might be optimal for the investor in our model not to buy any stock, even when the risk premium is positive. Further analysis of the optimal policy is also provided.

Reload options, call options granting new options on exercise, are popularly used in compensation. Although the compound option feature may seem complicated, there is a distribution-free dominant policy of exercising reload options whenever they are in the money. The optimal policy implies general formulas for numerical valuation. Simpler formulas for valuation and hedging follow from Black-Scholes assumptions with or without continuous dividends. Time vesting affects the optimal policy, but numerical results indicate that it is nearly optimal to exercise in the money whenever feasible. The results suggest that reload options produce similar incentives as employee stock options and share grants.

Financial transaction taxes, or generally transaction costs, are salient in derivatives markets and seldom studied in equilibrium models. We study a tractable model with proportional transaction costs where agents trade a derivative with nonlinear payoffs to hedge nontraded endowments. We show that trade is sustained in an equilibrium with transaction costs only if there is sufficient heterogeneity in risk aversion. When there is trade, the equilibrium return variance increases in transaction costs. These results are driven by how mean-variance demands, hedging demands, and asymmetry of no-transaction region widths determine the equilibrium Sharpe ratio and return volatility when transaction costs change.

[The Black-Scholes-Merton option valuation method involves deriving and solving a partial differential equation (PDE). But this method can generate multiple values for an option. We provide new solutions for the Cox-Ingersoll-Ross (CIR) term structure model, the constant elasticity of variance (CEV) model, and the Heston stochastic volatility model. Multiple solutions reflect asset pricing bubbles, dominated investments, and (possibly infeasible) arbitrages. We provide conditions to rule out bubbles on underlying prices. If they are not satisfied, put-call parity might not hold, American calls have no optimal exercise policy, and lookback calls have infinite value. We clarify a longstanding conjecture of Cox, Ingersoll, and Ross.]

We examine the optimal trading strategy for a CRRA investor who maximizes the expected utility of wealth on a finite date and faces transaction costs. Closed-form solutions are obtained when this date is uncertain. We then show a sequence of analytical solutions converge to the solution to the problem with a deterministic finite horizon. Consistent with the common life-cycle investment advice, the optimal trading strategy is found to be horizon dependent and largely buy and hold. Moreover, it might be optimal for the investor in our model not to buy any stock, even when the risk premium is positive. Further analysis of the optimal policy is also provided.

Reload options, call options granting new options on exercise, are popularly used in compensation. Although the compound option feature may seem complicated, there is a distribution-free dominant policy of exercising reload options whenever they are in the money. The optimal policy implies general formulas for numerical valuation. Simpler formulas for valuation and hedging follow from Black–Scholes assumptions with or without continuous dividends. Time vesting affects the optimal policy, but numerical results indicate that it is nearly optimal to exercise in the money whenever feasible. The results suggest that reload options produce similar incentives as employee stock options and share grants.

The Black-Scholes-Merton option valuation method involves deriving and solving a partial differential equation (PDE). But this method can generate multiple values for an option. We provide new solutions for the Cox-Ingersoll-Ross (CIR) term structure model, the constant elasticity of variance (CEV) model, and the Heston stochastic volatility model. Multiple solutions reflect asset pricing bubbles, dominated investments, and (possibly infeasible) arbitrages. We provide conditions to rule out bubbles on underlying prices. If they are not satisfied, put-call parity might not hold, American calls have no optimal exercise policy, and lookback calls have infinite value. We clarify a longstanding conjecture of Cox, Ingersoll, and Ross.